Modular Form Data by Ken-ichi SHIOTA

Summary

I have calculated the follwing data on the spaces M20(q)) of the elliptic modular forms of weight 2, for all prime levels 11 <= q < 10000:
• The (q,∞)-quaternion algebra D over Q
• A maximal order O in D
• The class number H and the type number T of D
• H is equal to the dimension of M20(q)).
• T is equal to the dimension of the minus-eigenspace M2-0(q)) of the Atkin-Lehner involution.
• The representatives { Aj } j = 1,...,H of the left O-ideal classes
• The maximal orders Oj = (Aj)-1Aj in D
• The representatives { Aij } i = 1,...,H of the left Oj-ideal classes
• Aij is obtained by Aij = ( a constant multiple of ) (Aj)-1Ai.
• The group order ej of the unit group of Oj
• The theta series θij associated to the norm form of Aij
• The definition of the theta series is
θij(z) = (1/ej) Σ x in Aij exp(2πiz N(x)/N(Aij))
where N denotes the reduced norm on D.
• We have ej θij = ei θji, hence we have to calculate θij only for i >= j.
• The Brandt matrices B(n) ( n = 0,1,2,... )
• The definition of the Brandt matrices is
Σ n = 0,1,2,... B(n) exp(2πinz) = ( θij(z) ) i,j = 1,...,H
• For n >= 1, B(n) is an integral representation matrix of the Hecke operator T(n) on M20(q)).
• The characteristic polynomial of B(n) ( = that of T(n) ) and its factorization over Z
Further, I am calculating the follwing data:
• The common eigenvectors of B(n)'s, and the eigenvalues of B(n0) for a selected n0
• Each eigenvector corresponds to the Eisenstein series E or a newform f in the space of the elliptic cusp forms S20(q)).
• Each eigenvalue is equal to a Hecke eigenvalue, hence to a Fourier coefficient of E or f.

Reference

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